In
probability and
statistics, an elliptical distribution is any member of a broad family of
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
s that generalize the
multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms an
ellipse and an
ellipsoid, respectively, in iso-density plots.
In
statistics, the normal distribution is used in ''classical''
multivariate analysis
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the diff ...
, while elliptical distributions are used in ''generalized'' multivariate analysis, for the study of symmetric distributions with tails that are
heavy
Heavy may refer to:
Measures
* Heavy (aeronautics), a term used by pilots and air traffic controllers to refer to aircraft capable of 300,000 lbs or more takeoff weight
* Heavy, a characterization of objects with substantial weight
* Heavy, ...
, like the
multivariate t-distribution, or light (in comparison with the normal distribution). Some statistical methods that were originally motivated by the study of the normal distribution have good performance for general elliptical distributions (with finite variance), particularly for spherical distributions (which are defined below). Elliptical distributions are also used in
robust statistics
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such ...
to evaluate proposed multivariate-statistical procedures.
Definition
Elliptical distributions are defined in terms of the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
of probability theory. A random vector
on a
Euclidean space has an ''elliptical distribution'' if its characteristic function
satisfies the following
functional equation (for every column-vector
)
:
for some
location parameter , some
nonnegative-definite matrix and some scalar function
.
The definition of elliptical distributions for ''real'' random-vectors has been extended to accommodate random vectors in Euclidean spaces over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
complex numbers, so facilitating applications in
time-series analysis
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
. Computational methods are available for generating
pseudo-random
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process.
Background
The generation of random numbers has many uses, such as for rando ...
vectors from elliptical distributions, for use in
Monte Carlo simulations for example.
Some elliptical distributions are alternatively defined in terms of their
density functions. An elliptical distribution with a density function ''f'' has the form:
:
where
is the
normalizing constant,
is an
-dimensional
random vector
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value ...
with
median vector (which is also the mean vector if the latter exists), and
is a
positive definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...
which is proportional to the
covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square Matrix (mathematics), matrix giving the covariance between ea ...
if the latter exists.
Examples
Examples include the following multivariate probability distributions:
*
Multivariate normal distribution
*
Multivariate ''t''-distribution
*
Symmetric multivariate stable distribution
*
Symmetric multivariate Laplace distribution
*
Multivariate logistic distribution
Multivariate may refer to:
In mathematics
* Multivariable calculus
* Multivariate function
* Multivariate polynomial
In computing
* Multivariate cryptography
* Multivariate division algorithm
* Multivariate interpolation
* Multivariate optical c ...
* Multivariate symmetric general
hyperbolic distribution
The hyperbolic distribution is a continuous probability distribution characterized by the logarithm of the probability density function being a hyperbola. Thus the distribution decreases exponentially, which is more slowly than the normal distrib ...
[
]
Properties
In the 2-dimensional case, if the density exists, each iso-density locus (the set of ''x''1,''x''2 pairs all giving a particular value of ) is an ellipse or a union of ellipses (hence the name elliptical distribution). More generally, for arbitrary ''n'', the iso-density loci are unions of ellipsoids. All these ellipsoids or ellipses have the common center μ and are scaled copies (homothets) of each other.
The multivariate normal distribution is the special case in which . While the multivariate normal is unbounded (each element of can take on arbitrarily large positive or negative values with non-zero probability, because for all non-negative ), in general elliptical distributions can be bounded or unbounded—such a distribution is bounded if for all greater than some value.
There exist elliptical distributions that have undefined mean, such as the Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) func ...
(even in the univariate case). Because the variable ''x'' enters the density function quadratically, all elliptical distributions are symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
about
If two subsets of a jointly elliptical random vector are uncorrelated
In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, ther ...
, then if their means exist they are mean independent of each other (the mean of each subvector conditional on the value of the other subvector equals the unconditional mean).
If random vector ''X'' is elliptically distributed, then so is ''DX'' for any matrix ''D'' with full row rank. Thus any linear combination of the components of ''X'' is elliptical (though not necessarily with the same elliptical distribution), and any subset of ''X'' is elliptical.
Applications
Elliptical distributions are used in statistics and in economics.
In mathematical economics, elliptical distributions have been used to describe portfolios in mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
.
Statistics: Generalized multivariate analysis
In statistics, the multivariate ''normal'' distribution (of Gauss) is used in ''classical'' multivariate analysis
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the diff ...
, in which most methods for estimation and hypothesis-testing are motivated for the normal distribution. In contrast to classical multivariate analysis, ''generalized'' multivariate analysis refers to research on elliptical distributions without the restriction of normality.
For suitable elliptical distributions, some classical methods continue to have good properties. Under finite-variance assumptions, an extension of Cochran's theorem In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance.
Statement
Let ''U''1, ..., ''U'N'' be i.i. ...
(on the distribution of quadratic forms) holds.
Spherical distribution
An elliptical distribution with a zero mean and variance in the form where is the identity-matrix is called a ''spherical distribution''. For spherical distributions, classical results on parameter-estimation and hypothesis-testing hold have been extended. Similar results hold for linear models
In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, the term ...
, and indeed also for complicated models ( especially for the growth curve model). The analysis of multivariate models uses multilinear algebra
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p ...
(particularly Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
s and vectorization
Vectorization may refer to:
Computing
* Array programming, a style of computer programming where operations are applied to whole arrays instead of individual elements
* Automatic vectorization, a compiler optimization that transforms loops to vect ...
) and matrix calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a mult ...
.
Robust statistics: Asymptotics
Another use of elliptical distributions is in robust statistics
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such ...
, in which researchers examine how statistical procedures perform on the class of elliptical distributions, to gain insight into the procedures' performance on even more general problems,[
] for example by using the limiting theory of statistics ("asymptotics").
Economics and finance
Elliptical distributions are important in portfolio theory
Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversificati ...
because, if the returns on all assets available for portfolio formation are jointly elliptically distributed, then all portfolios can be characterized completely by their location and scale – that is, any two portfolios with identical location and scale of portfolio return have identical distributions of portfolio return. Various features of portfolio analysis, including mutual fund separation theorem In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in a ...
s and the Capital Asset Pricing Model
In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio.
The model takes into accou ...
, hold for all elliptical distributions.
Notes
References
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*:Originally
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Further reading
* A collection of papers.
{{DEFAULTSORT:Elliptical Distribution
Types of probability distributions
Location-scale family probability distributions
Multivariate statistics
Normal distribution